Theory of the Trojan asteroids

In the previously published Parts I and II of the paper, the author has constructed a formal long-periodic solution for the case of 11 resonance in the restricted problem of three bodies to 0(m ^3/2), wherem is the small mass parameter of the system. The time-dependencet(, ,m), where is the mean synodic longitude and is related to the Jacobi constant, has been expressed by ahyperelliptic integral. It is shown here that with the approximationm=0 in the integrand, the functiont(, , 0) can be expanded in a series involving standardelliptic functions. Then the problem of inversion can be formally solved, yielding the function (t, , 0).Similarly, the normalized period (,m) of the motion can be approximated by theHagihara hyperelliptic integral (, 0), corresponding tom=0. This integral is also expanded into elliptic functions. Asymptotic forms for (, 0) are derived for 0 and for 1, corresponding to the extreme members of thetadpole branch of the family of orbits.     

A theory of the Trojan asteroids

The paper constructs a long-periodic solution for the case of 11 resonance in the restricted problem of three bodies. The solution is smoothed by the exclusion of the internal resonant terms arising from the near-commensurability between the long and the short periods of the asteroid.Although the Brown (1911) conjecture regarding the termination of the family of the tadpoles at the Lagrangian pointL ₃ is not supported by our analysis, the conjecture seems to hold in the limit asm0.     

Opposition effect of Trojan asteroids

CCD-photometry of three Jupiter Trojan asteroids were carried out to study their opposition effect. We obtained well-sampled magnitude–phase curves for (588) Achilles, (884) Priamus, and (1143) Odysseus in the maximal attainable phase angle range down to 0.1–0.2°. The magnitude–phase relations have a linear behavior in all observed range of phase angles and do not show any non-linear opposition brightening. We have not found any confident differences between phase slopes measured in B, V and R bands. The values of the measured phase slopes of Trojans are different from available data for Centaurs. They are within the range of phase slopes measured for some low-albedo main belt asteroids, also exhibit a linear behavior down to small phase angles. An absence of non-linear opposition brightening puts constraints on the surface properties of the studied objects, assuming very dark surfaces where single scattering plays dominating role.We also determined the rotation periods, amplitudes, the values of color indexes B–V and V–R, and the absolute magnitudes of these asteroids.     

Critical inclination of Trojan asteroids

The author's earlier solution for Trojan asteroids is developed further. It is shown that depending on the amplitude of libration around the Lagrangian point L₄, there is a critical inclination which determines the sign of the variation of the ascending node. If the orbital inclination of a Trojan is smaller than the critical one, then the ascending node decreases and otherwise it increases. The variation of the eccentricity and of the longitude of the perihelion has also a dependence on the critical inclination.     

Theory of the Trojan asteroids Part II

In the previously published (1977a) Part I of the paper, the author has constructed a formal long-periodic solution for the case of 1:1 resonance in the restricted problem of three bodies. Here the accuracy of the solution is carried fromO(m) toO(m ^3/2), wherem is the mass parameter of the system.Asymptotic approximations for the period of the motion are obtained for the case of small oscillations about the Lagrangian pointL ₄, in agreement with the classical theory, and for the vicinity of a logarithmic singularity on themean separatrix, passing throughL ₃. The regularizing function (), which removes the singularities of the Poincaré type, is extended to all orders, and the result is used to prove the periodicity of the solution.     

Theory of the Trojan asteroids,IV

In a previous publication (1977) the author has constructed a family () of long-periodic orbits in the Trojan case of the restricted problems of three bodies. Here he constructs the domain of the analytical solution of the problem of the motion, excluding the vicinity of thecritical divisor which vanishes at the exact commensurability of the natural frequencies ₁ and ₂. In terms of thecritical masses m_j(₂), or the associatedcritical energies _j ² (m), is the intersection of the intervals ofshallow resonance, of the form. Inasmuch as the intervals |₂– _j ² |<_j ofdeep resonance aredisjoint, it follows that (1) the disjointed family () embraces the tadpole branch, 0²1, lying in: and (2) despite the clustering of _j ² (m) atj=, the family () includes, for ²=1, an asymptoticseparatrix that terminates the branch in the vicinity of the Lagrangian pointL ₃.In a similar manner, the family () can be extended to the horseshoe branch 1<_{2 2} ² .     

Discovery and rediscovery of Trojan asteroids

During 1987–1994, observational campaigns with different telescopes at several observatories have been initiated by the author in order to discover new Trojans. The importance of Trojan asteroids comes from celestial mechanics, where they represent the physical solution of the famous Lagrange triangular problem. Their importance lies also in the fact, that they may have some relation with comets. Furthermore, the Trojan belt may be as large as the belt of asteroids. Moreover, recently families have been discovered between the already well known Trojans. Enough reasons to continue to search for these interesting objects.     

The three-dimensional motion of Trojan asteroids

The problem is considered within the framework of the elliptic restricted three-body problem. The asymptotic solution is derived by a three-variable expansion procedure. The variables of the expansion represent three time-scales of the asteroids: the revolution around the Sun, the libration around the triangular Lagrangian pointsL ₄,L ₅, and the motion of the perihelion. The solution is obtained completely in the first order and partly in the second order. The results are given in explicit form for the coordinates as functions of the true anomaly of Jupiter. As an example for the perturbations of the orbital elements the main perturbations of the eccentricity, the perihelion longitude and the longitude of the ascending node are given. Conditions for the libration of the perihelion are also discussed.     

Discovery and rediscovery of Trojan asteroids

During 1987–1994, observational campaigns with different telescopes at several observatories have been initiated by the author in order to discover new Trojans. The importance of Trojan asteroids comes from celestial mechanics, where they represent the physical solution of the famous Lagrange triangular problem. Their importance lies also in the fact, that they may have some relation with comets. Furthermore, the Trojan belt may be as large as the belt of asteroids. Moreover, recently “families” have been discovered between the already well known Trojans. Enough reasons to continue to search for these interesting objects.     

Long periodic perturbations of Trojan asteroids

The motion of the Trojan asteroids is studied in the elliptic restricted three-body problem of the Sun-Jupiter-asteroid system. Long periodic perturbations of the orbital elements are discussed. Relations between dynamical parameters are considered and comparisons are made with Bien's and Schubart's results.     

Recent progress in the theory of Trojan asteroids

In previous publications the author has constructed a long-periodic solution of the problem of the motion of the Trojan asteroids, treated as the case of 1:1 resonance in the restricted problem of three bodies. The recent progress reported here is summarized under three headings:

1. The nature on the long-periodic family of orbits is re-examined in the light of the results of the numerical integrations carried out by Deprit and Henrard (1970). In the vicinity of the critical divisor $$D_k \equiv \omega _1 - k\omega _2 ,$$ not accessible to our solution, the family is interrupted by bifurcations and shortperiodic bridges. Parametrized by the normalized Jacobi constant α², our family may, accordingly, be defined as the intersection of admissible intervals, in the form $$L = \mathop \cap \limits_j \left\{ {\left| {\alpha - \alpha _j } \right| > \varepsilon _j } \right\};j = k,k + 1, \ldots \infty .$$ Here, {α_j(m)} is the sequence of the critical α_j corresponding to the exactj: 1 commensurability between the characteristic frequencies ω₁ and ω₂ for a given value of the mass parameterm. Inasmuch as the ‘critical’ intervals |α?α_j|<ε_j can be shown to be disjoint, it follows that, despite the clustering of the sequence {α_j} at α=1, asj→∞, the family extends into the vicinity of the separatrix α=1, which terminates the ‘tadpole’ branch of the family.
2. Our analysis of the epicyclic terms of the solution, carrying the critical divisorD _k , supports the Deprit and Henrard refutation of the E. W. Brown conjecture (1911) regarding the termination of the tadpole branch at the Lagrangian pointL ₃. However, the conjecture may be revived in a refined form. “The separatrix α=1 of the tadpole branch spirals asymptotically toward a limit cycle centered onL ₃.”
3. The periodT(α,m) of the libration in the mean synodic longitude λ in the range $$\lambda _1 \leqslant \lambda \leqslant \lambda _2$$ is given by a hyperelliptic integral. This integral is formally expanded in a power series inm and α² or \(\beta \equiv \sqrt {1 - \alpha ^2 }\) .

The large amplitude of the libration, peculiar to our solution, is made possible by the mode of the expansion of the disturbing functionR. Rather than expanding about Lagrangian pointL ₄, with the coordinatesr=1, θ=π/3, we have expandedR about the circler=1. This procedure is equivalent to analytic continuation, for it replaces the circle of convergence centered atL ₄ by an annulus |r?1|<ε with 0≤θ<2π.     

The motion of the perihelion of Trojan asteroids

The problem is investigated by using the equations of Jupiter's main perturbations in the eccentricity and in the perihelion longitude of Trojan asteroids. The limits and the period of the variation of the eccentricity and of the perihelion longitude are calculated for 30 Trojans. The perihelion is shown to circulate in 20 cases and to librate for 10 asteroids.     

Physical properties and orbital stability of the Trojan asteroids

All the Trojan asteroids orbit about the Sun at roughly the same heliocentric distance as Jupiter. Differences in the observed visible reflection spectra range from neutral to red, with no ultra-red objects found so far. Given that the Trojan asteroids are collisionally evolved, a certain degree of variability is expected. Additionally, cosmic radiation and sublimation are important factors in modifying icy surfaces even at those large heliocentric distances. We search for correlations between physical and dynamical properties, we explore relationships between the following four quantities; the normalised visible reflectivity indexes (S^′), the absolute magnitudes, the observed albedos and the orbital stability of the Trojans. We present here visible spectroscopic spectra of 25 Trojans. This new data increase by a factor of about 5 the size of the sample of visible spectra of Jupiter Trojans on unstable orbits. The observations were carried out at the ESO-NTT telescope (3.5 m) at La Silla, Chile, the ING-WHT (4.2 m) and NOT (2.5 m) at Roque de los Muchachos observatory, La Palma, Spain. We have found a correlation between the size distribution and the orbital stability. The absolute-magnitude distribution of the Trojans in stable orbits is found to be bimodal, while the one of the unstable orbits is unimodal, with a slope similar to that of the small stable Trojans. This supports the hypothesis that the unstable objects are mainly byproducts of physical collisions. The values of S^′ of both the stable and the unstable Trojans are uniformly distributed over a wide range, from 0%/1000 Å to about 15%/1000 Å. The values for the stable Trojans tend to be slightly redder than the unstable ones, but no significant statistical difference is found.     

The theory of the Trojan asteroids,part V

E.W. Brown conjectured (1911) that the family of the long-periodic orbits in the Troian case of the restricted problem of three bodies terminates in an asymptotic orbit passing through the Lagrangian point L₃ at t=±∞. In 1977 the author showed that such an orbit deviates from L₃ by the epicyclic term mg (±∞). It is shown here that $$g\left( { \pm \infty } \right) = 0,$$ so that the Brown conjecture regarding L₃ is false. Contrary to what Brown believed, there is an entire family ofhomoclinic orbits, doubly asymptotic to short-periodic orbits around L₃. In the complex z-plane of the Poincaré eccentric variables, the latter orbits are circles of radius mR, with R bounded away from zero. The kinematics of the homoclinic family is investigated here in some detail.     

Albedos and diameters of three Mars Trojan asteroids

We observed the Mars Trojan Asteroids (5261) Eureka and (101429) 1998 VF₃₁ and the candidate Mars Trojan 2001 FR₁₂₇ at 11.2 and 18.1 microns using Michelle on the Gemini North telescope. We derive diameters of 1.28, 0.78, and <0.52 km, respectively, with corresponding geometric visible albedos of 0.39, 0.32, and >0.14. The albedos for Eureka and 1998 VF₃₁ are consistent with the taxonomic classes and compositions (S(I)/angritic and S(VII)/achondritic, respectively) and implied histories presented in a companion paper by Rivkin et al. Eureka's surface likely has a relatively high thermal inertia, implying a thin regolith that is consistent with predictions and the small size that we derive.     

On the origin of the asteroids

Initial temperature distribution of the nebulous disk around the sun led to the bordering region between the asteriods and Jupiter becoming a transition region, in which the “icy” matter changed from uncondensed to condensed state. This continuity in chemical constitution makes the projected surface density of solid condensate in the inner portion of the Jupiter region greater than the mean value in the asteriod region, and the time taken for dust particles to precipitate to become comparable to that in the asteriod region. Owing to gravitational instability, the dust layer then broke up into clusters of particles, each cluster transforming into a planetesimal. The mass of such a planetesimal and the rate of its growth in the Jupiter region are greater than those in the asteriod region. Perturbations and mutual encounters between the larger planetesimals in the inner portion of the Jupiter region changed their orbits and produced rather large relative velocities. Some of these entered the asteriod region, drew out most of the matter there, and increased the random velocity of the asteriods that remained, so that they could not combine to form a planet. We prove the above view by a quantitative analysis in this paper.     

Collisions, cosmic radiation and the colors of the Trojan asteroids

The Trojan asteroids orbit about the Lagrangian points of Jupiter and the residence times about their present location are very long for most of them. If these bodies originated in the outer Solar System, they should be mainly composed of water ice, but, in contrast with comets, all the volatiles close to the surface would have been lost long ago. Irrespective of the rotation period, and hence the surface temperature and ice sublimation rate, a dust layer exists always on the surface. We show that the timescale for resurfacing the entire surface of the Trojan asteroids is similar to that of the flattening of the red spectrum of the new dust by solar-proton irradiation. This, if the cut-off radius of the size distribution of the impacting objects is between 1 mm and 1 m and its slope is −3, for the entire size range. Therefore, the surfaces of most Trojan asteroids should be composed mainly of unirradiated dust.     

On some long time dynamical features of the Trojan asteroids of Jupiter

The equation of motion of long periodic libration around the Lagrangian point $L_4$ L 4 in the restricted three-body problem is investigated. The range of validity of an approximate analytical solution in the tadpole region is determined by numerical integration. The predictions of the model of libration are tested on the Trojan asteroids of Jupiter. The long time evolution of the orbital eccentricity and the longitude of the perihelion of the Trojan asteroids, under the effect of the four giant planets, is also investigated and a slight dynamical asymmetry is shown between the two groups of Trojans at $L_4$ L 4 and $L_5$ L 5 .     

Note on the number and origin of apollo asteroids

Fifteen Apollo, or Earth-crossing, asteroids are now recognized, and their orbital elements tabulated here. None has been accidentally rediscovered, a circumstance that leads to the 50% probability that the total number equals or does not exceed 100 to absolute magnitude 18, or above a diameter of roughly 1 km. Physical observations, orbital characteristics, and associated meteor streams provide clues as to whether the Apollo asteroids are truly minor planets or moribund cometary nuclei. A definitive answer as to the origin of the Apollo asteroids is yet to be found, but a short discussion of the status of the problem is presented.Paper dedicated to Prof. Harold C. Urey on the occasion of his 80th birthday on 29 April, 1973.